Geometric dilation of closed planar curves: New lower bounds

Given two points on a closed planar curve, C, we can divide the length of a shortest connecting path in C by their Euclidean distance. The supremum of these ratios, taken over all pairs of points on the curve, is called the geometric dilation of C. We provide lower bounds for the dilation of closed curves in terms of their geometric properties, and prove that the circle is the only closed curve achieving a dilation of π/2, which is the smallest dilation possible. Our main tool is a new geometric transformation technique based on the perimeter halving pairs of C.